Advance Analytics with R (UG 21-24)
I am Ayush.
I am a researcher working at the intersection of data, law, development and economics.
I teach Data Science using R at Gokhale Institute of Politics and Economics
I am a RStudio (Posit) certified tidyverse Instructor.
I am a Researcher at Oxford Poverty and Human development Initiative (OPHI), at the University of Oxford.
Reach me
ayush.ap58@gmail.com
ayush.patel@gipe.ac.in
Dip our toes into classification techniques. How to apply and assess these methods.
References for this lecture:
“….often the methods used for classification first predict the probability that the observation belongs to each of the categories of a qualitative variable, as the basis for making the classification. In this sense they also behave like regression methods.”
Default data| default | student | balance | income |
|---|---|---|---|
| No | No | 729.5265 | 44361.625 |
| No | Yes | 817.1804 | 12106.135 |
| No | No | 1073.5492 | 31767.139 |
| No | No | 529.2506 | 35704.494 |
| No | No | 785.6559 | 38463.496 |
| No | Yes | 919.5885 | 7491.559 |
| No | No | 825.5133 | 24905.227 |
| No | Yes | 808.6675 | 17600.451 |
| No | No | 1161.0579 | 37468.529 |
| No | No | 0.0000 | 29275.268 |
| No | Yes | 0.0000 | 21871.073 |
| No | Yes | 1220.5838 | 13268.562 |
| No | No | 237.0451 | 28251.695 |
| No | No | 606.7423 | 44994.556 |
| No | No | 1112.9684 | 23810.174 |
| No | No | 286.2326 | 45042.413 |
| No | No | 0.0000 | 50265.312 |
| No | Yes | 527.5402 | 17636.540 |
| No | No | 485.9369 | 61566.106 |
| No | No | 1095.0727 | 26464.631 |
Default is our response(\(Y\)).Yes or No.I ran this: \(p(balance) = \beta_0 + \beta_1X\)
## make a dummy for default
Default|>
mutate(
default_dumm = ifelse(
default == "Yes",
1,0
)
)-> def_dum
## regress dummy over balance and plot
lm(default_dumm ~ balance,
data = def_dum)|>
broom::augment()|>
ggplot(aes(balance,default_dumm))+
geom_point(alpha= 0.6)+
geom_line(aes(balance, .fitted),
colour = "red")+
labs(
title = "Linear regression fit to qualitative response",
subtitle = "Yes =1, No = 0",
y = "prob default status"
)+
theme_minimal() -> plot_linear
## Run the logistic regression
glm(
default_dumm ~ balance,
data = def_dum,
family = binomial
)|>
broom::augment(type.predict = "response")|>
ggplot(aes(balance,default_dumm))+
geom_point(alpha= 0.6)+
geom_line(aes(balance, .fitted),
colour = "red")+
labs(
title = "Logistic regression fit to qualitative response",
subtitle = "Yes =1, No = 0",
y = "prob default status"
)+
theme_minimal() -> logistic_plotWe saw that some fitted values in the linear model were negative.
We need a function that will return values between [0,1].
\[p(X) = \frac{e^{(\beta_0 + \beta_1X)}}{1+e^{\beta_0 + \beta_1X}}\]
This is the logistic function, modeled by the maximum likelihood method.
odds:
\[\frac{p(X)}{1-p(X)}\] **log odds or logit:
\[log(\frac{p(X)}{1-p(X)}) = \beta_0 + \beta_1X\]
if the following are the results of the model \(logit(p(default)) = \beta_0 + \beta_1Balance\):
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | -10.651330614 | 0.3611573721 | -29.49221 | 3.623124e-191 |
| balance | 0.005498917 | 0.0002203702 | 24.95309 | 1.976602e-137 |
What is the probability of default with balance $5000??
\[p(X) = \frac{e^{(\beta_0 + \beta_1X_1 + \beta_2X_2+...+\beta_nX_n)}}{1+e^{\beta_0 + \beta_1X_1 + \beta_2X_2+...+\beta_nX_n}}\]
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | -1.086905e+01 | 4.922555e-01 | -22.080088 | 4.911280e-108 |
| income | 3.033450e-06 | 8.202615e-06 | 0.369815 | 7.115203e-01 |
| balance | 5.736505e-03 | 2.318945e-04 | 24.737563 | 4.219578e-135 |
| studentYes | -6.467758e-01 | 2.362525e-01 | -2.737646 | 6.188063e-03 |
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | -3.5041278 | 0.07071301 | -49.554219 | 0.0000000000 |
| studentYes | 0.4048871 | 0.11501883 | 3.520181 | 0.0004312529 |
There is no consesus in statistics community over a single measure that can describe a goodness of fit for logistic regression.
Use the Credit data in {ISLR}.
What you just did is called Stratified binary model.
to Multinomial Logistic Regression
\[Pr(Y=k|X=x) = \frac{e^{\beta_{k0}+\beta_{k1}x_1+...+\beta_{kp}xp}}{1+\sum_{l=1}^{K-1}e^{\beta_{l0}+\beta_{l1}x_1+...+\beta_{lp}x_p}}\]
for k = 1,…K-1, and
\[Pr(Y=K|X=x) = \frac{1}{1+\sum_{l=1}^{K-1}e^{\beta_{l0}+\beta_{l1}x_1+...+\beta_{lp}x_p}}\]
\[log(\frac{Pr(Y=k|X=x)}{Pr(Y=K|X=x)}) = \beta_{k0}+\beta_{k1}x_1+...+\beta_{kp}xp\]
Which class is treated as reference or baseline is unimportant.
How to interpret this?
Rows: 344
Columns: 8
$ species <fct> Adelie, Adelie, Adelie, Adelie, Adelie, Adelie, Adel…
$ island <fct> Torgersen, Torgersen, Torgersen, Torgersen, Torgerse…
$ bill_length_mm <dbl> 39.1, 39.5, 40.3, NA, 36.7, 39.3, 38.9, 39.2, 34.1, …
$ bill_depth_mm <dbl> 18.7, 17.4, 18.0, NA, 19.3, 20.6, 17.8, 19.6, 18.1, …
$ flipper_length_mm <int> 181, 186, 195, NA, 193, 190, 181, 195, 193, 190, 186…
$ body_mass_g <int> 3750, 3800, 3250, NA, 3450, 3650, 3625, 4675, 3475, …
$ sex <fct> male, female, female, NA, female, male, female, male…
$ year <int> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007…
multi_log <- nnet::multinom(
formula = species ~ body_mass_g + bill_length_mm + bill_depth_mm + flipper_length_mm + sex + island,
data = peng_ref
)# weights: 27 (16 variable)
initial value 365.837892
iter 10 value 21.914358
iter 20 value 1.629266
iter 30 value 0.026372
final value 0.000049
converged
Call:
nnet::multinom(formula = species ~ body_mass_g + bill_length_mm +
bill_depth_mm + flipper_length_mm + sex + island, data = peng_ref)
Coefficients:
(Intercept) body_mass_g bill_length_mm bill_depth_mm
Adelie 502.6573 -0.08755830 -20.075027 34.82987
Chinstrap -434.3867 -0.02106537 6.332771 -16.48865
flipper_length_mm sexmale islandDream islandTorgersen
Adelie 0.5054518 33.23469 62.03886 144.9809
Chinstrap 1.7645190 -55.22699 335.85058 63.1425
Std. Errors:
(Intercept) body_mass_g bill_length_mm bill_depth_mm
Adelie 0.5314853 2.351402 29.93540 5.286822
Chinstrap 0.5310960 4.080649 29.91681 5.278463
flipper_length_mm sexmale islandDream islandTorgersen
Adelie 49.88305 0.2294146 0.531096 4.701009e-47
Chinstrap 49.81079 0.2290253 0.531096 4.261135e-130
Residual Deviance: 9.874339e-05
AIC: 32.0001
# calculate z-statistics of coefficients
z_stats <- summary(multi_log)$coefficients/
summary(multi_log)$standard.errors
# convert to p-values
p_values <- (1 - pnorm(abs(z_stats)))*2
# display p-values in transposed data frame
data.frame(t(p_values)) Adelie Chinstrap
(Intercept) 0.000000e+00 0.000000000
body_mass_g 9.702963e-01 0.995881131
bill_length_mm 5.024680e-01 0.832357200
bill_depth_mm 4.456258e-11 0.001785562
flipper_length_mm 9.919154e-01 0.971741303
sexmale 0.000000e+00 0.000000000
islandDream 0.000000e+00 0.000000000
islandTorgersen 0.000000e+00 0.000000000
Gentoo Adelie Chinstrap
1 1.565008e-135 1.000000e+00 1.009721e-242
2 3.833780e-97 1.000000e+00 1.450741e-166
3 3.913549e-122 1.000000e+00 1.006490e-181
5 3.854489e-165 1.000000e+00 2.652195e-247
6 2.628864e-168 1.000000e+00 9.671388e-281
7 5.558841e-114 1.000000e+00 3.782674e-190
8 3.576898e-116 1.000000e+00 2.880335e-227
13 3.717985e-108 1.000000e+00 3.470609e-172
14 5.313520e-178 1.000000e+00 2.906555e-297
15 4.397228e-190 1.000000e+00 9.358591e-320
16 4.638659e-132 1.000000e+00 3.570151e-212
17 1.323762e-143 1.000000e+00 1.383387e-216
18 3.906700e-111 1.000000e+00 6.765722e-218
19 2.342237e-174 1.000000e+00 3.742224e-262
20 1.803219e-103 1.000000e+00 6.877397e-206
21 3.440325e-75 1.000000e+00 1.084873e-188
22 2.942167e-90 1.000000e+00 4.087688e-228
23 1.885524e-93 1.000000e+00 8.693837e-211
24 1.304105e-64 1.000000e+00 3.625997e-196
25 2.258588e-50 1.000000e+00 2.714209e-176
26 1.050278e-93 1.000000e+00 4.472482e-212
27 5.070290e-66 1.000000e+00 1.977702e-192
28 4.657299e-56 1.000000e+00 1.773975e-147
29 6.353727e-88 1.000000e+00 1.524263e-202
30 1.459617e-55 1.000000e+00 2.364103e-190
31 1.084357e-69 1.000000e+00 9.173878e-17
32 1.227459e-100 1.000000e+00 5.420913e-94
33 1.265506e-86 1.000000e+00 2.282452e-34
34 8.494378e-82 1.000000e+00 4.163335e-66
35 3.892077e-102 1.000000e+00 1.530044e-47
36 5.034347e-123 1.000000e+00 7.437468e-121
37 3.676963e-116 1.000000e+00 5.542033e-110
38 2.071518e-62 1.000000e+00 3.696922e-16
39 2.421798e-124 1.000000e+00 2.037458e-93
40 6.809138e-66 1.000000e+00 1.601284e-61
41 3.425600e-120 1.000000e+00 7.624079e-81
42 1.606314e-77 1.000000e+00 4.276671e-50
43 6.806524e-135 1.000000e+00 5.590874e-97
44 1.276361e-49 1.000000e+00 3.090448e-26
45 1.481080e-105 1.000000e+00 2.763406e-53
46 2.564070e-66 1.000000e+00 2.715698e-55
47 3.442093e-99 1.000000e+00 7.482183e-81
49 2.187000e-113 1.000000e+00 2.595632e-71
50 2.061308e-96 1.000000e+00 2.896132e-90
51 2.979432e-49 1.000000e+00 3.169427e-145
52 1.697699e-47 1.000000e+00 1.852417e-180
53 3.668664e-95 1.000000e+00 1.072902e-201
54 3.787966e-52 1.000000e+00 1.067250e-172
55 8.398157e-123 1.000000e+00 2.068522e-229
56 4.242336e-55 1.000000e+00 1.503984e-174
57 1.477876e-49 1.000000e+00 3.319481e-146
58 9.797629e-62 1.000000e+00 3.358364e-184
59 2.927841e-83 1.000000e+00 9.109545e-178
60 1.502988e-94 1.000000e+00 3.440099e-226
61 3.045152e-84 1.000000e+00 8.887102e-183
62 4.785268e-68 1.000000e+00 5.169778e-209
63 4.444320e-52 1.000000e+00 3.202953e-150
64 1.419865e-38 1.000000e+00 2.020648e-158
65 2.360178e-92 1.000000e+00 2.038745e-188
66 5.421996e-35 1.000000e+00 4.069958e-151
67 5.476171e-70 1.000000e+00 3.160205e-158
68 2.078893e-49 1.000000e+00 3.188062e-179
69 7.954905e-146 1.000000e+00 1.710102e-209
70 1.077474e-99 1.000000e+00 7.562552e-198
71 3.865774e-182 1.000000e+00 1.261787e-274
72 4.918193e-122 1.000000e+00 6.676413e-220
73 6.012949e-105 1.000000e+00 1.034230e-162
74 1.910730e-68 1.000000e+00 4.867732e-150
75 3.284153e-138 1.000000e+00 2.276590e-215
76 2.618102e-84 1.000000e+00 9.775178e-174
77 9.229092e-81 1.000000e+00 2.732396e-137
78 1.219806e-157 1.000000e+00 3.841097e-274
79 5.636064e-116 1.000000e+00 4.127281e-182
80 5.403569e-109 1.000000e+00 2.345329e-202
81 2.595345e-160 1.000000e+00 5.445574e-234
82 6.249382e-53 1.000000e+00 5.461387e-139
83 5.962076e-143 1.000000e+00 2.474997e-229
84 1.620648e-166 1.000000e+00 1.214968e-284
85 1.460100e-104 1.000000e+00 1.627128e-56
86 5.435650e-115 1.000000e+00 2.320514e-97
87 4.252100e-136 1.000000e+00 7.652729e-132
88 5.233585e-114 1.000000e+00 1.076593e-75
89 3.364375e-108 1.000000e+00 1.960486e-98
90 5.170773e-96 1.000000e+00 8.844224e-54
91 2.399353e-116 1.000000e+00 1.564235e-67
92 2.369104e-57 1.000000e+00 5.984160e-23
93 1.586023e-119 1.000000e+00 1.344923e-80
94 1.484469e-60 1.000000e+00 3.282288e-46
95 1.300253e-107 1.000000e+00 1.283398e-61
96 9.985286e-73 1.000000e+00 1.402530e-42
97 3.686971e-96 1.000000e+00 1.308798e-55
98 1.683356e-66 1.000000e+00 1.620603e-44
99 1.008476e-129 1.000000e+00 6.726151e-87
100 7.608280e-50 1.000000e+00 1.154731e-20
101 3.825182e-85 1.000000e+00 1.162753e-192
102 2.022024e-43 1.000000e+00 3.536458e-174
103 1.322537e-55 1.000000e+00 4.846989e-143
104 1.577519e-86 1.000000e+00 1.054969e-231
105 4.338556e-101 1.000000e+00 1.474158e-197
106 1.261728e-78 1.000000e+00 6.837177e-209
107 9.369108e-44 1.000000e+00 3.189705e-131
108 2.378118e-96 1.000000e+00 3.188707e-237
109 5.296813e-63 1.000000e+00 6.022331e-159
110 6.780156e-06 9.999932e-01 1.699143e-128
111 1.872526e-34 1.000000e+00 9.761655e-120
112 5.672252e-10 1.000000e+00 2.800928e-138
113 2.520868e-61 1.000000e+00 6.490000e-149
114 3.439657e-41 1.000000e+00 1.510222e-165
115 1.613357e-80 1.000000e+00 8.393870e-198
116 4.589386e-26 1.000000e+00 2.167375e-139
117 1.333710e-133 1.000000e+00 7.219327e-193
118 1.875383e-181 1.000000e+00 6.421625e-293
119 5.416429e-142 1.000000e+00 1.382857e-212
120 1.686160e-134 1.000000e+00 1.870521e-225
121 7.970359e-148 1.000000e+00 3.536496e-218
122 1.292156e-177 1.000000e+00 3.222353e-281
123 4.198999e-96 1.000000e+00 3.384342e-165
124 2.604573e-112 1.000000e+00 8.559222e-197
125 1.837631e-140 1.000000e+00 4.157304e-204
126 1.947839e-121 1.000000e+00 3.828437e-215
127 2.478227e-127 1.000000e+00 1.775509e-191
128 1.024613e-90 1.000000e+00 9.595811e-183
129 1.396937e-126 1.000000e+00 1.546359e-184
130 3.280699e-78 1.000000e+00 1.061450e-146
131 2.298752e-132 1.000000e+00 1.046066e-200
132 3.065056e-121 1.000000e+00 1.841132e-206
133 3.030640e-114 1.000000e+00 2.000481e-72
134 8.112726e-87 1.000000e+00 2.224610e-71
135 7.842629e-91 1.000000e+00 6.644925e-43
136 3.409778e-60 1.000000e+00 2.490694e-29
137 1.683988e-121 1.000000e+00 2.224177e-74
138 1.033267e-106 1.000000e+00 5.770265e-90
139 2.701357e-84 1.000000e+00 8.079601e-33
140 8.450468e-66 1.000000e+00 1.488202e-42
141 3.152874e-67 9.999593e-01 4.068580e-05
142 1.617609e-75 1.000000e+00 2.721354e-42
143 7.409834e-126 1.000000e+00 3.397055e-75
144 8.994604e-63 1.000000e+00 7.923681e-28
145 5.783065e-103 1.000000e+00 8.678595e-44
146 4.605532e-105 1.000000e+00 4.108604e-90
147 4.342044e-80 1.000000e+00 1.572962e-65
148 1.885663e-113 1.000000e+00 3.916754e-78
149 5.687431e-113 1.000000e+00 2.382362e-66
150 2.105977e-104 1.000000e+00 3.060488e-83
151 4.036940e-91 1.000000e+00 6.654021e-48
152 1.912664e-70 1.000000e+00 3.972737e-38
153 1.000000e+00 1.772889e-109 1.371973e-36
154 1.000000e+00 1.293197e-123 1.826463e-68
155 1.000000e+00 7.520028e-117 3.422658e-36
156 1.000000e+00 6.893219e-143 8.765637e-70
157 1.000000e+00 8.385801e-122 6.314732e-71
158 1.000000e+00 1.507993e-110 7.335537e-39
159 1.000000e+00 1.320615e-93 2.768662e-51
160 1.000000e+00 2.262479e-93 3.099923e-74
161 1.000000e+00 1.120134e-78 2.435404e-46
162 1.000000e+00 1.043816e-91 2.769695e-77
163 1.000000e+00 1.266441e-61 1.520768e-53
164 1.000000e+00 2.726199e-115 3.866374e-79
165 1.000000e+00 9.945072e-102 6.813724e-41
166 1.000000e+00 8.140272e-145 4.315103e-75
167 1.000000e+00 1.696385e-74 1.841659e-45
168 1.000000e+00 3.811851e-135 1.988430e-77
169 1.000000e+00 4.187589e-56 1.407891e-47
170 1.000000e+00 4.529948e-158 3.567558e-75
171 1.000000e+00 1.564320e-102 6.698092e-50
172 1.000000e+00 7.030489e-119 2.242786e-66
173 1.000000e+00 3.026554e-158 8.663190e-63
174 1.000000e+00 3.137029e-99 3.705486e-50
175 1.000000e+00 4.648131e-89 2.375477e-42
176 1.000000e+00 1.702306e-77 1.311517e-80
177 1.000000e+00 3.163518e-101 3.495546e-46
178 1.000000e+00 3.054452e-88 1.327207e-76
180 1.000000e+00 1.724005e-125 3.039510e-76
181 1.000000e+00 3.604353e-115 2.263329e-40
182 1.000000e+00 3.118941e-135 1.353999e-67
183 1.000000e+00 7.598731e-100 9.617386e-71
184 1.000000e+00 1.264204e-73 3.452318e-56
185 1.000000e+00 6.903902e-103 9.568626e-57
186 1.000000e+00 4.939525e-210 2.816545e-50
187 1.000000e+00 3.582723e-134 7.603848e-39
188 1.000000e+00 1.878878e-100 8.760223e-78
189 1.000000e+00 4.506500e-88 2.221559e-52
190 1.000000e+00 5.736024e-45 2.434454e-95
191 1.000000e+00 4.502070e-80 3.721294e-46
192 1.000000e+00 7.073412e-113 2.117083e-81
193 1.000000e+00 5.134575e-52 8.682756e-47
194 1.000000e+00 9.195023e-126 3.007258e-71
195 1.000000e+00 1.487214e-87 2.630540e-41
196 1.000000e+00 9.689026e-107 2.712435e-62
197 1.000000e+00 4.463268e-130 5.531538e-69
198 1.000000e+00 5.495591e-91 1.539825e-47
199 1.000000e+00 1.805392e-82 2.836442e-41
200 1.000000e+00 1.028279e-123 2.594745e-65
201 1.000000e+00 7.028607e-120 1.460173e-44
202 1.000000e+00 2.064603e-77 6.387252e-86
203 1.000000e+00 3.070581e-112 2.416780e-46
204 1.000000e+00 8.443716e-131 7.699229e-61
205 1.000000e+00 5.034473e-79 8.759494e-48
206 1.000000e+00 1.247832e-118 2.619672e-56
207 1.000000e+00 1.046291e-108 3.826834e-43
208 1.000000e+00 4.093166e-71 1.731317e-77
209 1.000000e+00 6.860213e-72 2.136910e-48
210 1.000000e+00 1.269283e-79 8.616304e-73
211 1.000000e+00 2.750180e-63 1.025108e-55
212 1.000000e+00 7.667920e-138 1.981604e-63
213 1.000000e+00 1.118393e-82 1.219461e-42
214 1.000000e+00 1.993765e-98 3.966097e-72
215 1.000000e+00 6.105472e-91 1.731410e-39
216 1.000000e+00 4.647250e-168 4.056522e-51
217 1.000000e+00 1.385310e-97 2.832574e-40
218 1.000000e+00 2.621620e-114 1.352348e-72
220 1.000000e+00 8.632477e-125 8.344145e-71
221 1.000000e+00 2.591495e-77 7.813423e-46
222 1.000000e+00 3.248586e-145 3.191987e-61
223 1.000000e+00 1.311400e-104 1.558088e-43
224 1.000000e+00 3.566976e-78 7.591760e-74
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226 1.000000e+00 4.596155e-114 9.416621e-49
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